A framework \(ABCD\) of four uniform rods each of length \(a\) and weight \(w\) smoothly jointed together hangs in equilibrium under gravity with \(AB\) held in a horizontal position, \(B\) and \(D\) being joined by a light string of length \(b\) (\(< \sqrt{2}a\)). By the principle of Virtual Work, or otherwise, find the tension in the string.
A uniform chain of weight \(w\) per unit length hangs in equilibrium under gravity on a rough circular cylinder of radius \(a\) with a horizontal axis; the chain lies in a plane perpendicular to the axis, one end \(A\) is on the level of the axis, and the length of the chain exceeds \(\pi a\). If the chain is on the point of slipping and \(T\) is the tension at the point at an angular distance \(\theta\) (\(<\pi\)) from \(A\), establish the equation \[ \frac{d}{d\theta}(e^{-\mu\theta}T) = wae^{-\mu\theta}(\cos\theta+\mu\sin\theta), \] where \(\mu\) is the coefficient of friction, and find the length of the chain.
A solid is made by drilling a cylindrical hole of radius \(a\) from a uniform solid sphere of radius \(b\); the axis of the cylindrical part of the surface of the solid passes through the centre of the spherical part. Find a formula for the radius of gyration of the solid about its axis of symmetry, and show that the formula is correct in the limiting cases \(a=0, a=b\).
A sphere of mass \(m\) at rest on a horizontal table is struck by a second sphere of mass \(m\) which is moving on the table with velocity \(u\); the spheres are smooth and are of the same radius, and the coefficient of restitution between them is \(e\) (\(<1\)). Find the condition in which there is the maximum alteration in the direction of motion of the second sphere, and find the loss of kinetic energy in the impact in this case.
Two particles are projected under gravity from a point \(O\) with the same initial velocity in the same vertical plane through \(O\) at angles of elevation \(\alpha, \beta\). If the trajectories meet at the point \((h, k)\) referred to horizontal and upward vertical axes at \(O\), prove that \[ k=h\tan(\frac{1}{2}\alpha+\frac{1}{2}\beta-\frac{1}{2}\pi). \] By considering the limiting case of this result as \(\beta\to\alpha\), or otherwise, prove that to attain maximum range along a given straight line through \(O\) with a given initial velocity the direction of projection must bisect the angle between the line and the vertical through \(O\).
The driving force of a car is constant and the resisting forces vary as the square of its speed; the mass of the car is 1 ton, its maximum horse-power 42 and its maximum speed 75 miles per hour. Find the distance in which the car accelerates from 30 to 60 miles per hour.
A bead can slide freely on a straight wire \(AB\) of length \(l\) which is rotated in a horizontal plane with constant angular velocity \(\omega\) about its end \(A\). Initially the bead is projected along the wire with velocity \(V\) from \(A\). When the bead leaves the wire, what is the angle between the line of the wire and the direction of motion of the bead?
Two particles \(P_1, P_2\) of masses \(m_1, m_2\) are connected by a light elastic string of modulus \(\lambda\) and natural length \(l\) and lie at rest on a smooth horizontal table at a distance \(l\) apart. If an impulse \(I\) is applied to \(P_1\) in the direction \(P_2P_1\), prove that in the subsequent motion the greatest extension of the string is \[ I \sqrt{\frac{m_2 l}{m_1(m_1+m_2)\lambda}}, \] and find when it is first attained.
Show that a system of forces acting in a plane can be reduced to two forces of which one acts at a given point and the other acts in a given line. If referred to rectangular axes the original forces are \((X_1, Y_1), (X_2, Y_2), \dots, (X_n, Y_n)\) acting at \((x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\) respectively, effect the reduction
A uniform heavy rod \(AB\) of length \(2a\) is in equilibrium in a horizontal position in contact with a rough plane of coefficient of friction \(\mu\) inclined to the horizontal at an angle \(\alpha\). At the end \(A\) a gradually increasing force is applied up the plane in the direction of the line of greatest slope through \(A\). Show that when the rod begins to move it turns about a point at distance \[ a\left(2-\frac{2\tan\alpha}{\mu}\right)^{\frac{1}{2}} \] from \(A\).